Theorems of the alternative revisited

Abstract

Theorems of the alternative for linear algebraic equations and inequalities are considered in this paper. Classical theorems of the alternative, such as Gordan’s theorem, Farkas’ theorem, Stiemke’s theorem, Motzkin’s theorem, Slater’s theorem, Tucker’s theorem, Duffin’s theorem, Gale’s theorems, etc. are recalled. Theorems of this form are important for both linear algebra and mathematical programming, especially for mathematical programming problems with linear equality and/or inequality constraints. Some new theorems of the alternative are formulated and proved.

Subject Classification:

• (2010) 90C05
• 97H60

Keywords:

• Theorems of the alternative
• Linear programming
• Systems of linear equations and Inequalities